Primarily, my role in the department is to teach and nurture students in learning Mathematics. I have been fortunate to have worked within and outside India by choosing a career in Mathematics. I am happy to inspire young minds to pursue Mathematics as a career and for higher studies. Apart from classroom teaching I am willing to work with my students on various mathematical topics. I encourage my students to take part in different mathematical activities like attending seminars, participating in summer programmes, workshops and training sessions outside the formal university education. I have supervised internal and external students on various reading projects both during regular semesters and summer and winter breaks. I take part in outreach activities regularly, and I like to interact with students from different institutions.
Broadly my area of research is Complex Dynamics and Dynamical Systems. I am trying to continue my research apart from my teaching duties. For this I travel to various places to attend academic meetings or collaborate on a research topic.
I defended my PhD thesis at Université Toulouse III - Paul Sabatier, France on 5th February 2010. I was an EU Marie Curie Research Fellow during my PhD and I also received funding from CNRS and ANR. Then I worked at the International Centre for Theoretical Physics (ICTP), Italy as a visiting researcher from April 2010 to July 2010; where my fellowship was funded by ICTP, UNESCO & IAEA. Thereafter I joined Boston University, USA as a postdoctoral fellow and I worked there from September 2010 to August 2012. After that I moved to Paris and visited Université Paris-Est Créteil as 'chercheur invité' (invited researcher) from November 2012 to July 2013. Next I joined Harish-Chandra Research Institute in India as a postdoctoral fellow and worked there from August 2013 until December 2014; in this place my fellowship was funded by the Department of Atomic Energy, Govt of India. I started working at Presidency University as an Assistant Professor in January 2015.
Research Experience during PhD:
I worked on Circle Homeomorphisms and the shapes of Arnol'd tongues in my doctoral thesis. This is a topic where tools from Dynamical Systems of Real and Complex valued functions can be applied simultaneously. The study of circle homeomorphisms and rotation numbers were first introduced by Poincaré in 1885. The Arnol'd tongues were introduced by Arnol'd on his first paper on Small Denominators in 1961. Herman resolved a conjecture of Arnol'd and had several important results on this topic in 1970's. I had a result on the decay of the widths of the tongues under some specific condition and this is an improvement of Herman's work. I also had results on the boundaries of the Arnol'd tongues for which gave better dynamical insight.
Research after PhD:
1) (With Xavier Buff, Jordi Canela and Adam Epstein) We worked on the shape of the tips of the tongues for the double standard family.
2) (With Sampad Lahiry) We worked on the change of the lengths of the images of any smooth arc by iterates of the Hyperbolic Toral Automorphisms and Horseshoe maps. We then generalized this for diffeomorphisms on manifolds with a hyperbolic invariant set.
3) (With Subhamoy Mondal and Anubrato Bhattacharyya) We worked on the change of the total variation of a curve under iterates of a Devaney Chaotic self map on the real line, or a graph. We then studied the same problem in higher dimensions, especially on the closed unit square in the plane.
4) (With Anubrato Bhattacharyya, Sabyasachi Mukherjee) We worked on the simply connectedness of the tongues in the double standard family. We showed that the Hausdorff dimension of the maximal chaotic set on the circle varies real-analytically inside a tongue.
5) (With Anubrato Bhattacharyya) We considered a family of circle maps that are perturbations of the doubling map on the circle by a piecewise linear map. Among the hyperbolic components, we proved the existence of tongues and made experimental proof of eyes. We also showed how combinatorics of the attracting orbit determines if a given hyperbolic component is a tongue or an eye.
6) (With Krishnendu Gongopadhyay, Anubrato Bhattacharyya, Subhamoy Mondal) We studied the reversibility and strong reversibility of affine automorphisms of the two-torus. We also provided a simple geometric condition, based on Pick’s Theorem, that guarantees the existence of fixed points, along with a description of the dynamics of affine toral automorphisms. We computed the entropy and characterized when the conjugacy classes in the affine group are finite or uncountable.
Administrative Experience:
I have served in the following committees at Presidency University:
(1) Member of the departmental Board of Studies (2015-till date)
(2) Member of the departmental Examination Committee (2016-2018)
(3) Organizer of the Outreach Programme (2016-2019)
(4) Member of the GenEd Cell (2015-2017)
(5) Member of the Bicentennial Celebration Committee (2015-2017)
In the odd semester 2026 I am teaching the following papers:
1) Advanced Complex Analysis to second year MSc students/fourth year NEP students at PU.
2) Differential Calculus to first year BSc minor students at PU with Md. Nasim Akhtar.
3) MDC to first year NEP students at PU.
4) Ordinary Differential Equations to first year MSc/third year BSc students at IACS.
In the past semesters I have taught the following papers at Presidency University:
1) BSc honours students - Analysis I & II, Metric Spaces, Complex Analysis, Fourier Analysis, Real Analysis, Coordinate Geometry (2D & 3D), Algebra, Multivariate Calculus.
2) BSc GenEd/GE/MDC students - Linear Algebra, Real Analysis, Differential Calculus, Joy of Numbers I.
3) MSc students - Metric Topology, Complex Analysis, Discrete Dynamical Systems, Dynamical Systems.
In Boston University I taught the following:
- Undergraduate - Calculus for Life and Social Sciences, Linear Algebra for Math majors, Basic Linear Algebra for non Math majors, Differential Equations.
- Graduate - Introduction to the Dynamical Systems.
I have supervised summer students on different topics of Complex Analysis and Dynamical Systems. I have guided Purbayan Chakraborty for a year-long reading project on some Advanced Topics in Complex Analysis starting from the summer of 2017. Purbayan is a former MSc student at Presidency and he went on to join the M2 programme at Grenoble, France in 2018 after working with me. He completed his PhD in France in 2023. In the summer of 2019 I worked with Sampad Lahiry, who is a former student of IMA Bhubaneswar. Sampad joined TIFR CAM Bangalore centre in 2019 and presently he is a PhD student working for a Belgian-Australian joint research project. In 2020 I worked with Abhirup Basu from IIT Kanpur. During 2025-2026 I mentored Anindita Kanjilal on a reading project on Symbolic Dynamics, who started working with me just after her MSc from IISER TVM.
I am willing to work with visiting students on various topics of Dynamical Systems, Ergodic Theory, Complex Analysis and Complex Dynamics during summer and winter breaks. I am also interested in working on computational problems in Dynamical Systems which require coding. Prospective students are requested to contact by email.
Some important PG supervision of Presidency Students: I have supervised Partha Sarathi Ghosh for his Master's Project on "Linearization of Holomorphic Functions" in 2018. Rajibul Haque has done his Master's Project on Inverse Limit Space and Devaney Chaos in 2022-2023.
I have supervised reading projects of PG or PhD students on topics like Riemann Mapping theorem and the Carathéodory extension theorem, Topological Entropy and Chaos, Chaotic maps in one and two dimensions including Hyperbolic Toral Automorphisms, Conjugacy classes of linear toral automorphisms, Symbolic Dynamics, Quasiconformal mappings, Fourier Analysis, Differential Geometry etc.
PhD students:
1) Subhamoy Mondal: PhD started in 2019, thesis submitted in 2024, PhD awarded in 2025, first position after PhD - Assistant Professor, The Heritage College, Kolkata.
2) Anubrato Bhattacharyya: PhD started in 2021, thesis submitted in 2026, first position after PhD - Postdoctoral fellow, TIFR Mumbai.
3) Pradip Biswas: PhD started in 2025.
For prospective PhD students, I am willing to work on Dynamical Systems and Complex Dynamics assuming that the student has a strong background in Analysis on Metric Spaces and Differential Geometry. To work on Dynamical Systems any prospective student is advised to study the following texts: (1) "Introduction to Chaotic Dynamical Systems" by Robert L Devaney, (2) "Introduction to Dynamical Systems" by Michael Brin & Garrett Stuck. To work in Complex Dynamics one has to build a strong background in the advanced topics of Complex Analysis and for that one can study the book titled "Complex Analysis" by Elias M Stein & Rami Shakarchi. To learn the basics of Complex Dynamics one can study the following books: (1) "Iteration of Rational Functions: Complex Analytic Dynamical Systems" by Alan F Beardon, (2) "Complex Dynamics" by Lennart Carleson & Theodore Gamelin.
1. K. Banerjee, A. Bhattacharyya, S. Mukherjee, Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets, Nonlinearity, Vol 39, No 07, 2026. doi: https://doi.org/10.1088/1361-6544/ae8a05
2. S. Mondal, A. Bhattarchyya, K. Banerjee, Total variation and chaos on the unit square, Journal of Difference Equations and Applications, Vol 31, Issue 11, 2025, pp 1418–1430, doi: https://doi.org/10.1080/10236198.2025.2557950
3. K. Banerjee, A. Bhattacharyya, Paths of analytic circle diffeomorphisms, Geometric Function Theory and Related Topics, Contemporary Mathematics, Vol 820, American Mathematical Society, 2025, pp. 1-13., doi: https://doi.org/10.1090/conm/820/16406
4. K. Banerjee, A. Bhattacharyya, S. Mondal, Total Variation of a curve under Chaos on the Real line and on a finite graph, Qualitative Theory of Dynamical Systems, Volume 23, 16, 2024, 13 pages, doi: https://doi.org/10.1007/s12346-023-00871-3
5. K. Banerjee, X. Buff, J. Canela, A. Epstein, Tips of the tongues in the double standard family, Nonlinearity, Vol 34, No 12, 2021, pp 8174-8191, doi: https://doi.org/10.1088/1361-6544/ac2d80
6. K. Banerjee, Boundaries of the Arnol’d tongues and the standard family - preprint, arXiv:1402.4756, 30 pages.
7. K. Banerjee, On the widths of the Arnol’d Tongues, Ergodic Theory and Dynamical Systems, Vol 34, Issue 05, 2014, pp 1451-1463, doi: https://doi.org/10.1017/etds.2013.11
Presidency University Students Corner